Compact Proofs of Partial Knowledge for Overlapping CNF Formulae

At CRYPTO ’94, Cramer, Damgård, and Schoenmakers introduced a general technique for constructing honest-verifier zero-knowledge proofs of partial knowledge (PPK), where a prover Alice wants to prove to a verifier Bob she knows τ witnesses for τ claims out of k claims without revealing the indices of those τ claims. Their solution starts from a base honest-verifier zero-knowledge proof of knowledge Σ and requires to run in parallel k execution of the base protocol, giving a complexity of O(kγ(Σ)), where γ(Σ) is the communication complexity of the base protocol. However, modern practical scenarios require communication-efficient zero-knowledge proofs tailored to handle partial knowledge in specific application-dependent formats. In this paper, we propose a technique to compose a large class of Σ-protocols for atomic statements into Σ-protocols for PPK over formulae in conjunctive normal form (CNF) that overlap, in the sense that there is a common subset of literals among all clauses of the formula. In such formulae, the statement is expressed as a conjunction of m clauses, each of which consists of a disjunction of k literals (i.e., each literal is an atomic statement) and ℓ literals are shared among clauses. The prover, for a threshold parameter τ≤k, proves knowledge of at least τ witnesses for τ distinct literals in each clause. At the core of our protocol, there is a new technique to compose Σ-protocols for regular CNF relations (i.e., when τ=1) that exploits the overlap among clauses and that we then generalize to formulae where τ>1 providing improvements over state-of-the-art constructions.